Chasing the spin gap through the phase diagram of a frustrated Mott insulator

The quest for entangled spin excitations has stimulated intense research on frustrated magnetic systems. For almost two decades, the triangular-lattice Mott insulator κ-(BEDT-TTF)2Cu2(CN)3 has been one of the hottest candidates for a gapless quantum spin liquid with itinerant spinons. Very recently, however, this scenario was overturned as electron-spin-resonance (ESR) studies unveiled a spin gap, calling for reevaluation of the magnetic ground state. Here we achieve a precise mapping of this spin-gapped phase through the Mott transition by ultrahigh-resolution strain tuning. Our transport experiments reveal a reentrance of charge localization below T⋆ = 6 K associated with a gap size of 30–50 K. The negative slope of the insulator-metal boundary, dT⋆/dp < 0, evidences the low-entropy nature of the spin-singlet ground state. By tuning the enigmatic ‘6K anomaly’ through the phase diagram of κ-(BEDT-TTF)2Cu2(CN)3, we identify it as the transition to a valence-bond-solid phase, in agreement with previous thermal expansion and magnetic resonance studies. This spin-gapped insulating state persists at T → 0 until unconventional superconductivity and metallic transport proliferate.

Since Anderson's notion of resonating valence bonds 1 , the quest for quantum spin liquids (QSL) has been fueled by the idea that suppressing long-range antiferromagnetic (AFM) order can stabilize itinerant or even topological spin excitations 2,3 . What is often neglected is that the coupling of magnetic degrees of freedom to the lattice can result in valence-bond-solid (VBS) phases with paired electron spins. Such singlet states occur, for instance, in triangular-lattice organic compounds [4][5][6][7][8] and in form of the wellknown spin-Peierls (SP) states in quasi 1D systems 9,10 . Also in other frustrated materials the importance of valence-bond phases has been revived [11][12][13][14][15] . Such a VBS scenario is fully consistent with the report of a spin gap in the organic QSL candidate κ-(BEDT-TTF) 2 Cu 2 (CN) 3 that aroused great attention in the QSL community very recently 16,17 , in agreement with gapped excitations in thermal transport 18 and a phase transition at 6 K deduced from thermal expansion 19 and NQR measurements 20 . While this finding rules out a gapless scenario with mobile spinons, that would have explained the finite Sommerfeld coefficient in specific heat results 21 , it leaves open the possibility of gapped QSL phases 3,[22][23][24] .
The tunability of κ-(BEDT-TTF) 2 Cu 2 (CN) 3 -a pressure of 1.3 kbar triggers the transition from a Mott insulator to superconducting (SC) and metallic phases 5,6,25-28 as illustrated in Fig. 1d-provides the unique opportunity to scrutinize the entropy of its magnetic ground state. According to the Clausius-Clapeyron relation, the pressure-tuned metal-insulator transition (MIT) acquires a positive slope dT MI /dp > 0 in the T-p phase diagram between a paramagnetic Mott insulator with large spin entropy and a metal (Fig. 1b), reminiscent of the Pomeranchuk effect in 3 He 28 . On the other hand, the entropy of AFM and VBS states is smaller than that of the Fermi liquid (FL) metal, yielding negative slopes dT N /dp < 0 and dT ⋆ /dp < 0 of the MIT (Fig. 1a, c), respectively.
Here we utilize the slope of the insulator-metal boundary of κ-(BEDT-TTF) 2 Cu 2 (CN) 3 as a probe of the spin-gapped ground state below T ⋆ = 6 K, evidencing its low-entropy nature. For that, we map the low-temperature Mott MIT with ultrahigh precision by measuring dc transport upon a combination of biaxial compression and tensile strain. We find a reentrance of insulating behavior setting in below T ⋆ , which is successively suppressed as metallicity proliferates. Through comparison with previously reported structural 7,19 and magnetic 16 properties, our present results clearly identify the '6K anomaly' as the transition to a VBS phase with dT ⋆ /dp < 0 ( Fig. 1c) and, thus, we complete the phase diagram of κ-(BEDT-TTF) 2 Cu 2 (CN) 3 . Moreover, our transport data yield a gap size of 30-50 K for the spin-singlet state, which coincides with the unexpectedly large critical field of 60 T 17 .

Results
Among the QSL candidates, κ-(BEDT-TTF) 2 Cu 2 (CN) 3 has been studied most intensely as it exhibits not only frustrated AFM exchange interactions (J = 250 K 29 ), but also a paradigmatic Mott transition 25,28,30 . This layered triangular-lattice compound with t 0 =t close to unity 31,32 shows no AFM order 29 and, at the same time, it is one of the best solid-state realizations of the single-band Hubbard model 28 . As such, it provides experimental access to check the predictions of dynamical mean-field theory: while the Mott transition is first-order type below the critical endpoint T crit , a quantum-critical crossover at the quantum Widom line (QWL) occurs at T > T crit 28,30,33,34 . Moreover, applying 1-2 kbar pressure allows to probe unconventional SC 25,27 as well as charge transport in Fermi liquids and bad metals 35 . By combining the most recent findings in κ-(BEDT-TTF) 2 Cu 2 (CN) 3 28,30,35 and compounds with different degree of frustration 5,8,26,30,36-38 , in Fig. 1d we present the state-of-the-art phase diagram of dimerized organic Mott systems as a function of T, electronic correlations U/W and geometrical frustration t 0 =t.
In this study, we explore the phase space extremely close to the MIT, which requires fine tuning of the correlation strength with a resolution equivalent to 10 −2 kbar. While such high precision cannot be reached in conventional oil pressure cells 25,35 , also gas pressure experiments are limited at low temperatures due to solidification of helium 27,30 . To that end, we utilize strain transmitted through a substrate-a method previously applied in doping-tuned experiments 39 . As sketched in Fig. 2a, a single-crystalline κ-(BEDT-TTF) 2 Cu 2 (CN) 3 filament of 140 nm thickness is placed on a flexible polyethylene substrate at ambient conditions. Differential thermal expansion between sample and substrate (see Methods) causes an inplane biaxial compression ε biaxial equivalent to 1-2 kbar hydrostatic pressure, which is sufficient to push the sample across the Mott MIT (Fig. 2b). Our dc transport results in Fig. 2c yield metallic properties and an onset of SC below T c ≈ 4 K (ε c = 0, see inset). Note, uniaxial strain experiments on κ-(BEDT-TTF) 2 Cu 2 (CN) 3 revealed a considerable enhancement of T c and a broadened SC dome 40 than for isotropic compression. Here, biaxial strain does not affect the in-plane anisotropy and T c is comparable to hydrostatic pressure experiments 25,27 .
While differential thermal contraction tunes the sample across the Mott MIT, it does not allow to vary the correlation strength in arbitrary steps. In order to reach back into the Mott state in a quasi-continuous manner, we perform bandwidth-tuning via tensile strain ε c > 0 along the crystallographic c-axis. As indicated on the bottom of Fig. 2a, we control the correlation strength by varying the bending radius of the substrate, which yields a negative pressure of less than a kilobar. As such, uniaxial strain is significantly weaker than the overall pressure applied (|ε c | < |ε biaxial + ε c |), hence we do not expect major effects from a change in frustration t 0 =t-certainly less pronounced than in ref. 40. The blue shaded region in Fig. 2b indicates the phase space covered in our experiments with a fixed ε biaxial < 0 and varying tensile strain between ε c = 0 and 0.32%. The resulting transport curves in Fig. 2c exhibit a textbook Mott MIT: the resistivity maxima at the Brinkman-Rice temperature T BR are pushed to lower temperatures as correlations increase 25,35 . In addition, in the range between ε c = 0.17% and 0.25% we observe a distinct upturn of resistance upon cooling well , resulting in a positive slope dT MI /dp > 0 of the metal-insulator boundary, AFM order has much smaller entropy and the Clausius-Clapeyron relation yields dT N /dp < 0, as seen in κ-(BEDT-TTF) 2 Cu[N(CN) 2 ]Cl 36,37 . b No such 'reentrance' of insulating behavior is expected for a gapless QSL, possibly realized in triangular, kagome or honeycomb lattices 2,3 . c Similar to AFM, also the transition from a spin-gapped VBS insulator to a metal yields dT ⋆ /dp < 0, for instance in EtMe 3 P[Pd(dmit) 2 ] 2 5,6 . d Geometrical frustration t 0 =t controls the magnetic ground state of triangular-lattice Mott insulators, causing pronounced changes in the phase diagram affecting also unconventional superconductivity (SC). Experimentally, AFM has been observed for t 0 =t<1 in κ-(BEDT-TTF) 2 Cu[N(CN) 2 ]Cl and β 0 -[Pd(dmit) 2 ] 2 salts whereas magnetic order is absent in the QSL candidates κ-(BEDT-TTF) 2 Cu 2 (CN) 3 , κ-(BEDT-TTF) 2 Ag 2 (CN) 3 and EtMe 3 Sb[Pd(dmit) 2 ] 2 with t 0 =t close to unity 26,38 ; VBS states were reported for

Mott insulator G e o m e t r ic a l F r u s t r a t io n t '/ t
The frustration dependence of AFM and VBS phases is schematically indicated for these systems, calling for in-depth studies upon controlled variation of t 0 =t.
below T BR . The resistance minima (magenta arrows in Fig. 2c) shift up to T ⋆ = 6 K until metallic behavior is lost completely. Since this phenomenology lines up with the '6K anomaly' at ambient pressure, we identify it as the transition to a spin-gapped non-metallic state that gets suppressed at the Mott MIT. Note, a similar upturn of resistivity around the MIT has been recently observed in a pressure-dependent transport study on the sister compound κ-(BEDT-TTF) 2 Ag 2 (CN) 3 41 , which calls for further scrutiny.
The reentrant behavior at low temperatures is apparent not only from a sign change of dR/dT to a non-metallic slope (dR/dT < 0 below T ⋆ ), but can be seen also in the fully insulating curves ε c ≥ 0.27%. In Fig. 3 we plot the logarithmic derivative d ln R=d 1=T as a measure of the transport gap Δ(T). While the gap size is steadily reduced upon cooling from 100 K to 20 K, consistent with the temperaturedependent spectral weight shifts reported in previous optical studies 28,35 , we observe an upturn at low temperatures forming a peak around T ⋆ . For comparison, the VBS transition of EtMe 3 P[Pd(dmit) 2 ] 2 at T ⋆ = 24 K yields similar maxima in d ln R=d 1=T from which we estimate a ratio of gap size and transition temperature Δ VBS /(k B T ⋆ ) ≈ 5 5 . Assuming the same ratio in the title compound gives Δ VBS /k B ≈ 30 K for T ⋆ = 6 K, which compares well with the local maxima in the inset of Fig. 3.
In ambient pressure transport studies 28 , the regime T < 10 K remained inaccessible due to the extremely large resistance; here, in the vicinity of the MIT R(T) is small enough to be measured. Following its observation in thermodynamic and magnetic probes 16,19,21 , in our present work we identify the '6K anomaly' in the Mottinsulating state of κ-(BEDT-TTF) 2 Cu 2 (CN) 3 for the first time in charge transport as a local maximum of d ln R=d 1=T. Even more importantly, we obtain a quantitative estimate of the associated energy gap E g = 2Δ ≈ 30-50 K at T ⋆ . Note, this significantly exceeds the spin gap of 12 K estimated from ESR data 16 , which may indicate the involvement of other degrees of freedom (e.g., structural) in the transition in addition to magnetic interactions. Crucially, the gap size observed here lines up with the critical field of order 60 T estimated in ref. 17, which explains why magnetic fields of order 10 T have little effect on the transition 19,21 .
d ln R=d 1=T in the range T > 20 K reflects changes of the Mott-Hubbard gap. While the gap size reduces upon biaxial compression, it increases upon uniaxial tension; the negative values below T BR correspond to dR/dT > 0. Interestingly, the strain-induced change upon ε biaxial < 0 is approximately a rigid vertical shift of the ambient pressure data 28 by −130 K (gray dotted line in Fig. 3). Quantitatively, the effect of ε biaxial is approximately three times bigger than ε c = 0.32%, consistent with the red and blue arrows in Fig. 2b, respectively. In Fig. 4a, b we compare the structural changes at the '6K anomaly' in κ-(BEDT-TTF) 2 Cu 2 (CN) 3 with the VBS transition at T ⋆ = 24 K in EtMe 3 P[Pd(dmit) 2 ] 2 4 . Thermal expansion experiments by Manna et al. 7,19 revealed very similar anisotropic distortions within the conducting layers at T ⋆ in both quasi 2D organic compounds. As sketched in Fig. 4a, b, the crystal contracts along the c-axis and expands along the other in-plane direction upon cooling. Also other probes 20,42,43 observed pronounced anomalies at 6 K due to modifications of the crystal structure and possible symmetry breaking 17 , which are typical fingerprints of a VBS transition. In addition, recent ESR studies revealed a rapid drop of spin susceptibility χ s due to the opening of a spin gap 16 , which agrees with NMR Knight shift data from ref. 44 that yield χ s indistinguishable from zero for T → 0 17 , as shown in Fig. 4c.  3 was placed on a flexible PET substrate. Cooling from 300 K down to T → 0 imposes biaxial compression of order 1-2 kbar to the sample due to differential thermal contraction, yielding metallic and superconducting (SC) properties (red arrow in b). Uniaxial tensile strain ε c along the c-axis is applied via bending of the substrate in order to tune from the Fermi liquid (FL) through the Mott MIT (blue arrow in b). b Crossover and transition temperatures from refs. 27,30,35, as indicated. c Resistance of the biaxially compressed crystal for different ε c . Inset: SC below T c ≈ 4 K for ε c = 0 is evident in the log-log plot. Upon approaching the Mott state, SC is lost and the resistivity maxima at T BR shift to lower temperatures. In addition, our data reveal a reentrance of insulating behavior (magenta arrows) below T ⋆ < T BR . Altogether, these findings provide solid evidence for anisotropic structural changes at the transition to a singlet state with a gap in the spectrum of spin excitations. In addition, T ⋆ = 6 K at p = 0 directly interpolates to the reentrance of the MIT in our strain-dependent transport data, which is mapped in the false-color plot of d ln R=d 1=T in Fig. 4d. We find that the resulting phase diagram prominently displays dT ⋆ /dp < 0 as sketched in Fig. 1c, analog to the VBS Mott insulator EtMe 3 P[Pd(dmit) 2 ] 2 5,6 with similar thermal expansion anomalies (Fig. 4a, b). In light of these overwhelming experimental similarities, we conclude that at ambient pressure the enigmatic '6K anomaly' in κ-(BEDT-TTF) 2 Cu 2 (CN) 3 is the transition from a paramagnetic Mott insulator to a spin-gapped VBS state. Upon reducing the correlation strength, the VBS phase is suppressed and can be entered from a metallic state upon cooling, as observed here for 0.17% ≤ ε c ≤ 0.25%. As seen in Fig. 4d, in the limit T → 0 the nonmagnetic insulator is adjacent to the Cooper-paired state below T c ≈ 4 K, hence the '6K anomaly' may affect unconventional SC.

Discussion
Since the first reports of the '6K anomaly' and the absence of magnetic order 21,29 , there have been different interpretations of the experimental results acquired over two decades in independent groups [16][17][18][19]44 . Here, we put together now all these pieces and obtain a comprehensive understanding of the low-temperature physics of κ-(BEDT-TTF) 2 Cu 2 (CN) 3 . Our present ultrahigh-resolution tuning of the '6K anomaly' through the Mott MIT delivers the missing part required to establish it as a true phase transition to a low-entropy spin-singlet ground state with distorted crystal structure. In view of the very shape of the phase diagram presented here, the magnetic, thermal transport, thermodynamic and structural properties of κ-(BEDT-TTF) 2 Cu 2 (CN) 3 16,18-20,29,43,44 are consistent with a spin-gapped VBS state, essentially identical to that reported in other layered organic Mott systems [4][5][6][7][8] . While chiral QSL states with a spin gap [22][23][24] cannot be ruled out at present, it is unclear whether the distinct lattice effects of the '6K anomaly' 19 are compatible with these theoretical models. In absence of a 'smoking-gun' experiment deciding between the two possibilities, the principle of Occam's Razor 45 suggests to favor not the exotic (chiral QSL) but rather the established scenario (VBS), not least as the latter has already been observed in related triangular-lattice systems [4][5][6][7][8] .
As such, we point out that the spin-gapped low-entropy phase associated with the '6K anomaly' is not intrinsic to pure Mott insulators, but rather results from magneto-structural instabilities. Likely, the elastic energy involved in the lattice distortion at T ⋆ contributes to the unexpectedly large energy gap and critical field, exceeding the spin gap deduced from ESR data 16 and k B T ⋆ /μ B ≈ 9 T by far 17 . Previous works have suggested the involvement of charge degrees of freedom 19,20,43 , but a sizeable charge disproportionation has been ruled out by vibrational spectroscopy 46 . The recent observation of reentrant insulating behavior in the related QSL candidate κ-(BEDT-TTF) 2 Ag 2 (CN) 3 possibly provides another test bed to investigate these issues 41 .
Finally, we suggest that, in absence of the VBS, the metallic state would persist much deeper into the insulating region of the phase diagram-in Fig. 2b T FL and T BR extrapolate to zero well below p = 1 kbar 27,35 . To that end, elucidating the genuine Mott MIT in the limit T → 0 requires to suppress T ⋆ in κ-(BEDT-TTF) 2 Cu 2 (CN) 3 , potentially by varying t 0 =t through uniaxial strain (Fig. 1d) or by applying high magnetic fields B ≈ 50-100 T. To conclude, our findings highlight the importance to gain understanding and control of magneto-elastic coupling in frustrated spin systems, in particular in other QSL candidates.

Sample preparation and application of uniaxial bending strain
We prepared polyethylene terephthalate (PET) substrates (Teflex® FT7, Toyobo Film Solutions Limited) and patterned 18-nm-thick Au electrodes using photo-lithography. Thin single crystals of κ-(BEDT-TTF) 2 Cu 2 (CN) 3 were synthesized with electrolysis of a 1,1,2-trichloroethane [10% (v/v) ethanol] solution (50 ml), in which BEDT-TTF (20 mg), KCN (60mg), CuCN (40 mg), and 18-crown-6 (100 mg) are dissolved. We applied a current of 8 μA overnight and obtained tiny thin crystals of κ-(BEDT-TTF) 2 Cu 2 (CN) 3 . The crystals were pipetted to 2-propanol, and were manipulated with a tip of hair and put on the substrate. After the substrate was taken out from 2-propanol and dried, the thin crystal tightly adhered to the substrate.  Fig. 1c) originating from the '6K anomaly' at p = 0. Altogether, the ambient pressure data in b, c and dT ⋆ /dp < 0 in d provide solid evidence for a low-entropy spin-singlet phase with structural anisotropy --the hallmarks of a VBS.
As described in previous work 39,47,48 , cooling yields differential thermal expansion between sample and substrate: the length change is ≤1% between 0 K and 300 K for κ-(BEDT-TTF) 2 Cu 2 (CN) 3 19,49 compared to >>1% for PET. As a result, the sample is subject to biaxial compression of order 1-2 kbar, deduced from the transport results in Fig. 2 in comparison to previously published resistivity data 27,30,35 . We applied uniaxial bending strains by pushing the back of the substrate with a nanopositioner (ANPz51, attocube systems) as shown in Fig. 2a. Assuming that the bent substrate is an arc of a circle, the strain S is estimated as S = 4tx/(l 2 + 4x 2 ) 47,48 using the small angle approximation, where x is the displacement of the nanopositioner, t = 177 μm and l = 12 mm are the thickness and length of the substrate, respectively. The uniaxial tensile strain was applied at 100 K in descending order from 0% to 0.32%. The cooling rate of the sample was 0.5 K/min. The thickness of the studied κ-(BEDT-TTF) 2 Cu 2 (CN) 3 crystal (140 nm) was measured a posteriori using a step profiler. Throughout the manuscript we follow the convention that a reduction of the sample volume (upon compressive stress or pressure) corresponds to negative strain ε < 0, whereas an expansion of the sample size (upon tensile stress or 'negative' pressure) yields ε > 0. By comparing our results to transport measurements under hydrostatic pressure 30,35 , we estimate that the total applied strain ε biaxial + ε c corresponds to a hydrostatic pressure p = 1.55 kbar for ε c = 0, and p = 1.05 kbar for ε c = 0.32%, yielding a conversion factor Δε c /Δp ≈ − 0.6 %/kbar. The range (1.05-1.55 kbar) covered by tuning of ε c in Figs. 2c and 4d is indicated by the blue rectangle in Fig. 2b.

Data analysis
The QWL in Fig. 2b was taken from ref. 28 whereas T BR , T c , and T FL are from ref. 35 Fig. 4c were taken from ref. 44 and the Knight shift K outer was determined for the outer 13 C nuclei using a chemical shift σ = 117, as described in ref. 17. In the same panel, the ESR susceptibility χ s for magnetic field perpendicular to the layers (B∥a similar to the Knight shift data) was taken from Fig. 2a in ref. 16. Both χ s and K outer were normalized to their value at 10 K.
At elevated temperatures the resistivity maxima indicate T BR 35 . At lower temperatures in the phase coexistence regime T ≤ T crit ≈ 15 K 50 , strictly speaking, the resistivity maxima indicate percolation of metallic regions. We assign the sharp jumps in the resistance data in Fig. 2c, also seen as narrow peaks between 10-15 K in Fig. 3, to an artefact of the sample mounting. As such, the apparent strain dependence is naturally related to the volume change of the sample that interferes with the applied strain. A micro-crack will open or close for a particular differential thermal expansion between sample and substrate, reached at a different temperature for different strain conditions upon changing ε c .